Clinical statistics for non-statisticians: Day one

Start with a bad joke

Two statistics are sitting in a bar. One turns to the other and asks, “So, how do you like married life?”

The other statistic responds …

Put your reaction (“Ha ha”, “Groan”, etc.) in the chat box.

Before I begin anything important, I like to start with a silly joke. Now on Zoom, I often miss student reactions. So when I say something funny, I want you to type “Ha ha” or “Smile” or “LMFAO”. The acronym LMFAO means laughing my something … I forget how the rest of it goes.

Now if the joke is corny, like a really really bad pun, it’s okay to put “Groan”. The only thing bad is if I tell a joke and get no reaction at all.

I’ll be sneaking in some jokes throughout the talk and I really want a reaction from you, good or bad. If I don’t get any reaction to a bad pun, your “pun”ishment will be more bad puns.

So here’s the joke. It has been floating around on the Internet for quite a while, and I can’t find the person who gets credit for this. But here goes.

[Read joke and finish with] “It’s okay but you lose a degree of freedom.”

Okay, I’m waiting for reactions.

Introduction

Tell us one interesting number about yourself
Examples
- 8: I have traveled to eight countries outside the United States
  - (Canada, Italy, China, France, Russia, England, Holland, and Iceland)
- 29: I did not learn how to drive until I was 29 years old
- 1802: My highest chess rating was 1802, but I am not that good any more.

Your turn

A bit more about myself

PhD in Statistics in 1982 from the University of Iowa
Currently full professor
Part-time statistical consultant
Funded on 18 research grants
Over 100 peer-reviewed publications
Website with over 2,000 pages
Many invitations to talk at conferences

I have a PhD in Statistics from the University of Iowa. I have always had a strong interest in the computational side of Statistics. My dissertation was 150 pages, and 100 of those pages were computer generated graphs.

I am currently a full professor at the University of Missouri-Kansas City in the Department of Biomedical and Health Informatics. I also do statistical consulting on a part-time basis.

I have been a prolific researcher, receiving support from 18 different grants, and writing over 100 peer-reviewed publications.

I started a website in 1998, writing about data analysis, research ethics, and evidence based medicine. I wrote about two or three pages every week and my site now has over 2,000 pages. It shows the value of persistence.

I love to talk about Statistics and have given many presentations at regional, national, and international conferences. This ranges from short 15 minute talks to day long short courses.

Outline of the three day course

Day one: Numerical summaries and data visualization
Day two: Hypothesis testing and sampling
Day three: Statistical tests to compare treatment to a control and regression models

My goal: help you to become a better consumer of statistics

Day one topics

Numerical summaries
- When should you present the mean versus the median
- When should you present the range versus standard deviation
- How should you display percentages
- Why should you round liberally

Day one topics (continued)

Data visualization
- How should you display continuous data
- Why is the normal bell-shaped curve important
- How should you display categorical data
- How do you illustrate trends and patterns
- What are some common mistakes in the choice of colors

Quiz questions

Counting and proportions

Counts are the most common statistic
- Counts are error prone
- Counts require a solid operational definition

Student exercise

Count the number of occurrences of the letter “e”.

A quality control  program is easiest
to implement from the top down. 
Make sure that you understand the 
the commitment of time and money
that is involved. Every workplace is
different, but think about allocating
10% of your time and 10% of the 
time of all your employees to 
quality control.

Counting sperm

Figure 1: Image of a haemocytometer

Speaker notes

This image is take from the WHO laboratory manual for the examination and processing of human semen, published in 2021. It shows a haemocytometer, an instrument used for counting the number of cells. To get a proper count, you need to include any cells inside the four by four grid of large squares in the middle of this micrograph. But what does “inside” mean? Should you count only those cells entirely inside the four by four grid. Or should you include cells that are partially inside the grid?

One rule is to count cells if the head of the sperm cell touches the top or right side of a square, but not if it touches the bottom or left side of the square. And don’t count a sperm cell if only the tail is inside the square.

That’s not the only way you can do this, but just make sure that whatever convention you use for deciding “inside” versus “outside” is consistent across your laboratory.

Tables of counts, using the Titanic data.

Figure 2: Counts of survival by gender

Percentages dividing by column totals

Figure 3: Column percentages

Percentages dividing by row totals

Row percentages

Percentages divided by grand total

Cell percentages

My recommendations

Treatment or exposure as rows
Outcome as columns
Usually report row percentages
- Female mortality rate: 33%
- Male mortality rate: 83%
But sometimes column percentages
- Survivors: 68% female, 32% male

Some rationale for these choices

My way

               Survived
               No          Yes
Sex Female     33% (154)   67% (308)
    Male       83% (863)   17% (142)

Not my way

               Sex
               Female      Male
Survived  No   33% (154)   83% (863)
          Yes  67% (308)   17% (142)

On your own

Calculate row and column percentages for the following tables. Interpret your results.

Figure 4: Titanic passenger class counts

The mean (average)

Figure 6: Cartoon image of Professor Mean

The median

Figure 7: Road with a median strip

Calculation of the mean and median

Mean
- Add up all the values, divide by the sample size
Median
- Sort the data
  - Select the middle value if n is odd
  - go halfway between the two middle values if n is even

Formal mathematical definitions

Mean
- \(\bar{X}=\frac{1}{n}\Sigma X_i\)
Median
- Sorted values \(X_{[1]},X_{[2]},...,X_{[n]}\)
  - \(X_{[(n+1)/2]}\) if n is odd,
  - \((X_{[n/2]}+X_{[n/2+1]})/2\) if n is even

Bacteria before and after A/C upgrade

Room Before  After Change
 121   11.8   10.1   -1.7
 125    7.1    3.8   -3.3
 163    8.2    7.2   -1.0
 218   10.1   10.5    0.4
 233   10.8    8.3   -2.5
 264   14     12     -2.0
 324   14.6   12.1   -2.5
 325   14     13.7   -0.3

Before remediation mean

11.8 + 7.1 + 8.2 + 10.1 + 10.8 + 14 + 14.6 + 14 = 90.6

90.6 / 8 = 11.325

Round to 11.3

After remediation mean

10.1 + 3.8 + 7.2 + 10.5 + 8.3 + 12 + 12.1 + 13.7 = 77.7

77.7 / 8 = 9.7125

Round to 9.7

Before remediation median (1/4)

Before remediation median (2/4)

Before remediation median (3/4)

125   7.1  
  
163   8.2  
  
218  10.1  
  
233  10.8  10.8
  
121  11.8  11.8
  
264  14.0  
  
325  14.0  
  
324  14.6

Before remediation median (4/4)

125   7.1  
  
163   8.2  
  
218  10.1  
  
233  10.8  10.8
                  (10.8 + 11.8) / 2 = 11.3
121  11.8  11.8
  
264  14.0  
  
325  14.0  
  
324  14.6

After remediation median (1/4)

After remediation median (2/4)

After remediation median (3/4)

125   3.8  
  
163   7.2  
  
233   8.3  
  
121  10.1  10.1
  
218  10.5  10.5
  
264  12.0  
  
324  12.1  
  
325  13.7

After remediation median (4/4)

125   3.8  
  
163   7.2  
  
233   8.3  
  
121  10.1  10.1
                  (10.1 + 10.5) / 2 = 10.3
218  10.5  10.5
  
264  12.0  
  
324  12.1  
  
325  13.7

Choosing between the mean and median

When do you use the mean?
- When totals are important
When do you use the median
- When outliers/skewness might distort your conclusions
Often, either is fine

Criticisms of the mean and median

Are you combining apples and onions?
Are you ignoring minorities?

There’s a wonderful cartoon by Dana Fradon that appeared in The New Yorker in 1976. She shows a road going into town and the sign by the side of the road reads “Hillsdale, Founded 1802, Altitude 600, Population 3,700. Total 6,122.” You can’t add these things together.

It’s similar for means. There was a dataset showing housing prices for homes in Boston and none of the analyses seemed to make sense. The problem in Boston is that a small number of the houses had prices that were out of sync with their other homes. These were historical houses, such as Paul Revere’s house.

When you are averaging numbers, maybe it’s okay to have a few oranges in with the apples. A mix of apples and oranges is just fruit salad. You shouldn’t have a problem with that.

When it becomes a problem is when the data are so diverse that it becomes a mix of apples and onions. There are lots of great recipes that mix apples and oranges, but none that mix apples and onions.

The other problem is that an average may be a reasonable number to represent the majority of patients in your sample, but it may masks some important trends that appear in a minority.

This is a big problem in a larger context than just the mean or median. There are some very fancy high tech predition models that work very well for most people and the statistics like the mean and median back this up quite nicely. But the prediction models perform terribly for minority groups.

Use of the mean for ordinal data

Gould 1985

Figure 8: Gould 1985

Bridge 2001, PMID: 11405531

Figure 9: Bridge and McKenzie 2001

Bridge 2001, PMID: 11405531 (continued)

The measurement of airway resistance by the interrupter technique (Rint) needs standardization. Should measurements be made be during the expiratory or inspiratory phase of tidal breathing? In reported studies, the measurement of Rint has been calculated as the median or mean of a small number of values, is there an important difference?

Bridge 2001, PMID: 11405531 (continued)

In the present data the mean of a set of values contributing to a measurement was not significantly different from the median. However, the use of the median has been recommended since it is less affected by possible outlying values such as might be included by fully automated equipment.

Tosato 2021, PMID: 34352201

Figure 10: Tosato et al 2021

Tosato 2021, PMID: 34352201 (continued)

Symptom persistence weeks after laboratory-confirmed severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) clearance is a relatively common long-term complication of Coronavirus disease 2019 (COVID-19). Little is known about this phenomenon in older adults. The present study aimed at determining the prevalence of persistent symptoms among older COVID-19 survivors and identifying symptom patterns.

Tosato 2021, PMID: 34352201 (continued)

The mean age was 73.1 ± 6.2 years (median 72, interquartile range 27), and 63 (38.4%) were women. The average time elapsed from hospital discharge was 76.8 ± 20.3 days (range 25-109 days).

Ielapi 2021, PMID: 34968328

Figure 11: Tosato et al 2021

Ielapi 2021, PMID: 34968328 (continued)

Background. Insomnia is one of the major health problems related with a decrease in quality of life (QOL) and also in poor functioning in night-shift nurses, that also may negatively affect patients’ care. The aim of this study is to evaluate the prevalence of insomnia in night shift nurses.

Ielapi 2021, PMID: 34968328 (continued)

Excerpt from Table 1.

Data reported as mean ± standard deviation or median [Q1-Q3]

Overall (n = 2′355)
Age, years  40.4 ± 10.3
Months of work 168 [72–300]
Night shifts per month, number  6.3 ± 1.4
Time to reach workplace, minutes    45 [45–65]
Rest time, minutes  180 [4–240]
Rest in the afternoon, minutes  30 [0–120]
Number of coffees, mean 2.5 ± 1.5
Number of coffees during night shift, mean  1.4 ± 1.1

Chen 2019, PMID: 31806195

Figure 12: Chen et al 2019

Chen 2019, PMID: 31806195 (continued)

Background: The prices of newly approved cancer drugs have risen over the past decades. A key policy question is whether the clinical gains offered by these drugs in treating specific cancer indications justify the price increases.

Chen 2019, PMID: 31806195 (continued)

Results: We found that between 1995 and 2012, price increases outstripped median survival gains, a finding consistent with previous literature. Nevertheless, price per mean life-year gained increased at a considerably slower rate, suggesting that new drugs have been more effective in achieving longer-term survival. Between 2013 and 2017, price increases reflected equally large gains in median and mean survival, resulting in a flat profile for benefit-adjusted launch prices in recent years.

Percentiles

Figure 13: Illustration of the 75th percentile

Computing percentiles

Many formulas
- Differences are not worth fighting over
My preference (pth quantile)
- Sort the data
- Calculate p*(n+1)
- Is it a whole number?
  - Yes: Select that value, otherwise
  - No: Go halfway between
  - Special cases: p(n+1) < 1 or > n

Speaker notes

There are close to a dozen different ways to compute a percentile, but the differences between the values selected are small and not worth fussing about.

Here is my preference for choosing the pth quantile (remember that for quantiles, you range between 0 and 1, not between 0 and 100).

Calculate the quantity p*(n+1). If that value is a whole number, great! You just select that value. If it is a fractional value, round up and down and go halfway between.

Once in a while, you’ll get an extreme case, where p(n+1) is less than 1 or greater than n. Just use a bit of common sense.

If you have nine values and p(n+1) is 9.2, you can’t go halfway between the 9th and 10th observations. There is no 10th observation. So just choose the 9th or largest value.

Likewise if p(n+1) is 0.8, you can’t go halfway between the zeroth and first observation. There is no zeroth observation. Just choose the first or smallest value.

Some examples of percentile calculations

Example for n=39
- For 5th percentile, p(n+1)=2 -> 2nd smallest value
- For 4th percentile, p(n+1)=1.6 -> halfway between two smallest values
- For 2nd percentile, p(n+1)=0.8 -> smallest value

Some terminology

Percentile: goes from 0% to 100%
Quantile: goes from 0.0 to 1.0
- 90th percentile = 0.9 quantile
Quartiles: 25th, 50th, and 75th percentiles
- Lower quartile: 25th percentile
- Upper quartile: 75th percentile

Before remediation upper quartile (1/4)

Before remediation upper quartile (2/4)

Before remediation upper quartile (3/4)

125   7.1  
  
163   8.2  
  
218  10.1  
  
233  10.8  
  
121  11.8  
  
264  14.0  14
  
325  14.0  14
  
324  14.6

Before remediation upper quartile (4/4)

125   7.1  
  
163   8.2  
  
218  10.1  
  
233  10.8  
  
121  11.8  
  
264  14.0  14
                  (14 + 14) / 2 = 14
325  14.0  14
  
324  14.6

After remediation upper quartile (1/4)

After remediation upper quartile (2/4)

After remediation upper quartile (3/4)

125   3.8  
  
163   7.2  
  
233   8.3  
  
121  10.1  
  
218  10.5  
  
264  12.0  12
  
324  12.1  12.1
  
325  13.7

After remediation upper quartile (4/4)

125   3.8  
  
163   7.2  
  
233   8.3  
  
121  10.1  
  
218  10.5  
  
264  12.0  12
                  (12 + 12.1) / 2 = 12.05
324  12.1  12.1
  
325  13.7

When you should use percentiles

Characterize variation
- Middle 50% of the data
Exposure issues
- Not enough to control median exposure level
Quantify extremes
- What does “upper class” mean?
Quality control
- Almost all products must meet a minimum standard

Speaker notes

There are many reasons why you might be interested in percentiles rather than the mean or median. Actually, the median is a percentile, the 50th percentile, but what I mean is percentiles other than 50%.

One important use of percentiles is looking at the middle 50% of the data. This is the data between the lower quartile (25th percentile) and the upper quartile (75th percentile). Is the middle 50% of the data bunched tightly together or spread widely apart?

Percentiles are also important in the study of exposures. If you work in an environment where the median worker has a safe level of exposure, you could easily end up with 20%, 30% or more of the workers dying from unsafe exposures. It is important to insure that not just the median, but a very high percentile like the 99th percentile of exposure levels is at a safe level.

Percentiles also help to define extreme groups. You can, for example, define the term upper class as anyone earning more than the 90th percentile of income.

Percentiles also can help with quality control. If you make a claim about a product, you want to make sure that that claim is not valid at a median level but at a much higher level. You don’t sell 500 mg bottles of liquid Tylenol is your factory is churning out a median fill level of 500 mg. Half of your customers would be cheated. Instead you insure that the 98th percentile coming out of the factory floor is at least 500 mg. You lose a bit of money because most bottles contain more than 500 mg, but the cost of an irate customer is worth more than the cost of 50 overfilled bottles.

Standard deviation

\[S = \sqrt{\frac{1}{n-1}\Sigma(X_i-\bar{X})^2}\]

At least one alternative formulas.

Why is variation important

Variation = Noise
- Too much noise can hide signals
Variation = Heterogeneity
- Too little heterogeneity, hard to generalize
- Too much heterogeneity, mixing apples and oranges
Variation = Unpredictability
- Too much unpredictability, hard to prepare for the future
Variation = Risk
- Too much risk can create a financial burden

Speaker notes

I want to discuss measures of variation now. Variation gets at the heart and soul of clinical statistics. A large portion of statistical analysis involves characterizing variation.

Variation can be thought of as a measure of noise. In general, but not always, noise is bad. Consider measuring a patient’s glucose level, to see if you have early evidence of diabetes. Your glucose level varies a lot during the day based on whether you skipped breakfast or decided to get a mid-afternoon Snickers bar. Your glucose level is noisy. A high level might or might not mean trouble. A low value might or might not mean you are safe. The large standard deviation of your measures of blood glucose indicates noise.

That’s why you are asked to take an overnight fast before testing your blood glucose level. Controlling your diet by not eating anything after midnight provides a more consistent measure of blood glucose. It has a smaller standard deviation and a high or low value is more helpful in diagnosis.

Variation can also be thought of as a measure of heterogeneity. Heterogeneity is also bad sometimes, but there are times when you want a fair amount of heterogeneity. A research study that has a lot of variation is better at providing a complete picture of what a typical patient is. Outcomes that are consistent in the presence of demographic heterogeneity give you more confidence in generalizing the results of a research study. You have some assurance that the therapy is not restricted to helping a small segment of patients.

Too much heterogeneity, though, can mean that any summary measure is a mixture of apples and oranges. You have to find the right balance.

Variation can be equated to unpredictability. The number of beds needed in a hospital does vary, and this makes it difficult to staff properly. The more variation in beds needed, the more headaches you have.

Variation can also be equated to risk. If you invest in a new drug, paying millions or even billions of dollars in testing, you are doing so with the hope that your investment will pay off. Unfortunately, the market for your drug is uncertain, and you might end up with no market at all if your clinical trials fail to convince FDA. There is variation in the return on your investment, and the more variation there is, the more risky your development plans are.

Should you try to minimize variation?

Yes, for early studies
- Easier to detect signals
- Proof of concept trials
No, for later studies
- Easier to generalize results
- Pragmatic trials

Speaker notes

It is a bit of a generalization, but most researchers try to avoid variation in early studies. By early studies, I mean studies of therapies that have not yet been extensively tested in a broad range of settings. Less variation means that there is a greater chance to detect signals. You remove variation by using very strict entry criteria on who can get into the study. You remove variation by tightly controlling what the patient is allowed to do (e.g., no concommitant medications). You remove variation by tightly standardizing the delivery of the intervention and the assessment of the outcome. You reduce variation by removing patients who deviate from the research protocol requirements.

These are known as proof of concept trials. If a new therapy cannot succeed even under the tight controls, there is no point in studying it futher. But success in a tightly controlled environment does not guarantee success in the real world.

If you are planning a trial that comes after many similar trials, you actually may want to encourage variation. Broaden the inclusion criteria so that the patients in the trial look no different than the patients you see every day in your clinic.

Standard deviation

\[S = \sqrt{\frac{1}{n-1}\Sigma(X_i-\bar{X})^2}\]

At least one alternative formulas.

The bell shaped curve

Does your variation follow a bell shaped curve?
- Values in the middle are most common
- Frequencies taper off away from the center
- Symmetry on either side
A bell shaped curve = better characterization of variation

Not a bell shaped curve (1/4)

Figure 14: Bimodal histogram

Not a bell shaped curve (2/4)

Figure 15: Skewed histogram

Not a bell shaped curve (3/4)

Figure 16: Uniform histogram

Not a bell shaped curve (4/4)

Figure 17: Heavy-tailed histogram

A bell shaped curve (finally!)

Figure 18: Bell-shaped histogram

Plus or minus one standard deviation

Figure 19: Percentage within one s

Plus or minus two standard deviations

Figure 20: Percentage within two s

Plus or minus three standard deviations

Figure 21: Percentage within three s

Lin 2022, PMID: 36126916

Figure 22: Lin et al 2022

Lin et al 2022 patient ages

Figure 23: Excerpt from Table 1 of Lin et al 2022: ages

Lin et al 2022 Charlson Comorbidity Index

Figure 24: Excerpt from Table 1 of Lin et al 2022: CCI

Lin et al 2022 PHQ-2 scores

Figure 25: Excerpt from Table 1 of Lin et al 2022: PHQ-2

Visualization

Proportions

Bar chart

Error bars

Box plot

Histogram

Plot all the data

Rug plot

Which visualization to choose?

Tables versus graphs

How should you display continuous data

How should you display categorical data

How do you illustrate trends and patterns

What are some common mistakes in the choice of colors

http://www.pmean.com/posts/misuse-of-gradient/

http://blog.pmean.com/rainbows/

Primary colors

Figure 26: Color combinations

The RGB color system

#rrggbb format
- #000000 is pure black
- #FFFFFF is pure white
- #FF0000 is pure red
- #00FF00 is pure green
- #0000FF is pure blue
You can mix and match to get 16,777,216 colors
- #800080 is purple, #FF69B4 is pink, #40E0D0 is turquoise

Color combinations: yellow

Red plus green Red plus blue

Color combinations: magenta

Figure 27: Green plus blue

The color cube

Figure 28: Illustration of the color cube

The color cylinder

Figure 29: Color cylinder

Rainbow

Harsh contrasts

Lighter rainbow

## Darker rainbow

Gentler contrasts

Equally spaced hues

Figure 30: Color choices for nominal data

The rainbow gradient

Figure 31: Illustration of the rainbow gradient